What is an early reference for the fact that if a compact, connected $n$-manifold $M$ is covered by two open sets homeomorphic to $\mathbb{R}^n$ then $M$ is homeomorphic to $S^n$?

And is it true that if $M$ is a compact, connected $n$-manifold with boundary, and if $M$ is covered by two open sets homeomorphic to $\lbrace(x_1,\ldots,x_n) \in \mathbb{R}^n | x_n \ge 0\rbrace$, then $M$ is a closed ball?

Lee: I do not see how one could prove this without the annulus theorem, so the earliest reference (say in dimension at least 5) would be Kirby and Siebenmann; in dimension 4 Freedman and Quinn; in dimension 3 and lower I am not sure whom to attribute this result to.
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MishaDec 28 '12 at 23:38

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The second one follows from the first and the Schoenflies theorem by doubling.
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Ian AgolDec 28 '12 at 23:52

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@Agol: do you mean the Brown-Mazur theorem, i.e. the collared version of the Schoenflies theorem?
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Lee MosherDec 29 '12 at 0:03

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Sorry, I meant Newman in 1966 who proved the topological Poincare conjecture (or Smale, depending on which category you're interested in). Clearly your manifold is a homotopy sphere; the question is whether it was identified to be a sphere earlier than the proofs of the Poincare conjecture?
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Ian AgolDec 29 '12 at 0:53

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I think this problem has some relation to Ljusternik-Schnirelmann category number. Ljusternik-Schnirelmann category number is the minimal number of coordinate that can cover the manifold. In dimension two, if a manifold is not S^2 that the category number is 3. I know there is a theorem that the minimal critical point of a function over manifold is bigger than the category number. I don't know whether these two number is equal in other dimension.
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Siqi HeDec 29 '12 at 9:09

1 Answer
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I'll only discuss the first question (EDIT : Actually, I address the second question at the end). As Agol pointed out in the comments, for $n \geq 5$ this is an easy consequence of Newman's 1966 proof of the Poincare conjecture in the topological category.

I don't know if it was explicitly stated earlier than this. However, it can easily be derived from the main result of the paper

Brown's theorem is as follows. Assume that $M$ is a topological $n$-manifold and that for all compact $K \subset M$, there exists some open set $U \subset M$ with $K \subset U$ and $U \cong \mathbb{R}^n$. Then $M \cong \mathbb{R}^n$. Brown's proof is clever, but completely elementary.

To get the desired result from this, assume that $X = U_1 \cup U_2$ with $U_i \cong \mathbb{R}^n$ and that $X$ is compact. Let $\phi : \mathbb{R}^n \rightarrow U_1$ be a homeomorphism. It is enough to prove that $X \setminus \{\phi(0)\} \cong \mathbb{R}^n$. We will do this with Brown's theorem. Consider a compact set $K \subset X \setminus \{\phi(0)\}$. To verify Brown's criteria, it is enough to construct a homeomorphism $\psi : X \setminus \{\phi(0)\} \rightarrow X \setminus \{\phi(0)\}$ such that $\psi(K) \subset U_2$.

EDIT : Lee suggested that this might be able to address his second question too. I thought a bit about it, and I believe that it can. The key is the following "relative" version of Brown's theorem, which can be proven exactly like Brown's theorem.

Theorem : Let $(M,N)$ be a pair consisting of a topological $n$-manifold $M$ and a closed submanifold $N \subset M$. Assume that for all compact $K \subset M$, there exists some open set $U \subset M$ such that $K \subset U$ and such that the pair $(U,U \cap N)$ is homeomorphic to the pair $(\mathbb{R}^n,\mathbb{R}^{n-1})$ (the second embedded in the standard way). Then $(M,N) \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$.

To apply this, assume that $X$ is a compact manifold with boundary and that $X = U_1 \cup U_2$ with $(U_i,\partial U_i) \cong (\mathbb{R}^n_{\geq 0},\mathbb{R}^{n-1})$. Double $X$ to get a closed manifold $Y$, and let $Y' \subset Y$ be the image of the boundary of $X$. The open sets $U_i$ double to give an open cover $Y = V_1 \cup V_2$. Letting $V_i' = V_i \cap Y'$, we have $(V_i,V_i') \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$. Let $(M,M')$ be the result of deleting the image of $0$ in $(V_1,V_1')$. It is enough to prove that $(M,M') \cong (\mathbb{R}^n,\mathbb{R}^{n-1})$, and this can be proven just like above.

Of course, Agol answered the second question first -- it follows from the topological Schonfleiss theorem applied to the double, which was proven by Brown in